A US academic has devised a formula for what every football fan instinctively understands - referees' decisions tend to favour the team playing at home.

A study at Harvard University looked at 5,000 English Premier League matches involving 50 different referees.

Researcher Ryan Boyko came up with the equation that for every extra 10,000 people in the crowd, the advantage for the home team increases by 0.1 goals.

His study showed certain referees were more likely to be swayed by the crowd.

The potential is there for a game to be altered because of factors that subconsciously affect the referee

Ryan Boyko

The figures revealed that away teams scored fewer goals and gave away more penalties, findings which seem to imply that referees are making calls in favour of the home team, possibly as a result of the influence of the crowd.

Some individual referees proved to be more susceptible to crowd influence than others, but Mr Boyko would not name which ones.

More experienced referees were shown to be less biased by the impact of a large crowd, which suggests they may develop a resistance to the home fans' cries.

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Mr Boyko, a research assistant in the university's psychology department, said: "Individual referees and the size of the crowd present are variables that affect the home field advantage.

"In order to ensure that all games are equally fair, ideally, all referees should be equally unaffected by the spectators.